Abstract
We are concerned with a parabolic system derived from the complex valued nonlinear heat equation partial derivative(t)z = Delta z+z(p); t > 0, x is an element of R-N; with initial data z(0) = u(0)+iv(0); where p > 1 is an integer. We study the global existence and the large time behavior of solutions for data u(0)(x) similar to c vertical bar x vertical bar(-2 alpha 1), v(0)(x) similar to c vertical bar x vertical bar(-2 alpha'1) as vertical bar x vertical bar -> infinity (vertical bar c vertical bar is sufficiently small) such that max(alpha(1); alpha'(1)) < N/2; alpha(1) >= 1/(p - 1) and alpha'(1) >= 1/(p - 1) if p is odd, alpha'(1) >= (1+alpha(1))/p if p is even. Since we may take different decay rates for the real part and imaginary part of the initial data, we obtain asymptotic behaviors which cannot occur for the real-valued nonlinear heat equation. Also, these asymptotic behaviors depend on the parity of the power of the nonlinearity.